Suppose we have a portfolio of $n$ credits. In order the estimate the Portfolio Value at Risk (99,9) we use a standard vasicek model with the Ability to pay variable $A_i=\sqrt{\rho}x+\sqrt{1-\rho}z_i$ where $x$ is some systematic factor. From this we can derive the conditional probability of default given some realisation of $x$. Now as far as I understand, $x$ is drawn from some distribution which is the Monte Carlo Simulation, and than the Portfolio loss is calculated. This is done for $k$ runs, the portfolio losses are aggregated and we draw the VaR from this distribution (please correct me if I am wrong).

However, most models also model the correlation of each credits default which would be an $n \times n$ matrix. I do not understand how this matrix is applied to a model as the above, or any other for that matter.



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